Regularization of the Inverse Laplace Transform with Applications in Nuclear Magnetic Resonance Relaxometry Candidacy Exam

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1 Regularization of the Inverse Laplace Transform with Applications in Nuclear Magnetic Resonance Relaxometry Candidacy Exam Applied Mathematics, Applied Statistics, & Scientific Computation University of Maryland, College Park Advisors: John J. Benedetto, Alfredo Nava-Tudela Mathematics, IPST Mentor: Richard Spencer Laboratory of Clinical Investigations, NIA

2 Outline Nuclear Magnetic Resonance (NMR) Relaxometry Background Objective Motivation from Celik 1D Discrete Model 2D Model Extension Ordinary Least Squares Regularization Regularized Least Squares Methods to Choose the Regularization Parameter Hansen s L-Curve L-Curve as an Analytical Tool FINDCORNER Algorithm Problem Extensions

3 Nuclear Magnetic Resonance (NMR) Relaxometry Figure: Clockwise from top left: a. Local magnetization M emerges from alignment with magnetic field B 0. b. With an RF pulse, M aligns with the magnetic field B 1 in the transversal plane. c. After the pulse, M begins to realign with B 0. d. Components M lon (t) and M tr (t), characterized by decay rates T 1 and T 2, respectively, describe M(t) at time t. Images courtesy of Alfredo Nava-Tudela.

4 Objective A 1-dimensional continuous NMR relaxometry signal takes the form y(t) = 0 f (T 2 )e t/t 2 dt 2 + n(t) (1) where T 2 is the transversal decay rate, f (T 2 ) corresponds to the amplitude of the associated component, and n(t) is additive noise. Objective: Recover the distribution of amplitudes f (T 2 ) present in the signal via an inverse Laplace transform (ILT).

5 Toy Example Consider a signal y(t) = 0.6e t/t 2, e t/t 2,2 + n(t) (2) where the exact distribution f (T 2 ) is f (T 2 ) = 0.6 δ T2,1 (T 2 ) δ T2,2 (T 2 ) (3) The recovery of f (T 2 ) is unstable due to the sensitivity of the inversion to noise.

6 Motivation Celik et al [1] demonstrated stabilization of the ILT through the introduction of a second, indirect dimension. Figure: The experimental process applied by Celik [1]. The 2D ILT path illustrated by the solid arrows produced better resolution of peaks in a sparse signal than the 1D ILT path (dashed arrow).

7 Motivation Figure: The experimental process applied by Celik [1]. Inversions from 12 different noise realizations are shown, demonstrating the stability of the 2D ILT. Top: 1D ILT. Bottom: 2D ILT projection.

8 Discrete Model A 1D NMR signal takes discrete form z(t i ) = K F(T 2,j )e t i /T 2,j (4) j=1

9 Discrete Model A 1D NMR signal takes discrete form z(t i ) = K F(T 2,j )e t i /T 2,j (4) j=1 In matrix form: z = AF (5) where [A] ij = e t i /T 2,j (6) z i = z(t i ) (7)

10 2D NMR Model The 2D continuous NMR relaxometry signal takes the form z( t, t) = In discrete form, z( t i, t j ) = 0 0 K 1 K 2 k=1 m=1 F(T 1, T 2 )e t/t 1 e t/t 2 dt 1 dt 2 (8) F(T 1,k, T 2,m )e t i /T 1,k e t j /T 2,m (9)

11 2D NMR Model Define A 1, A 2, F, and Z such that [A 1 ] ik = e t i /T 1,k [ F]km = F(T 1,k, T 2,m) [A 2 ] jm = e t j /T 2,m [Z ] ij = z( t i, t j ) (10)

12 2D NMR Model Define A 1, A 2, F, and Z such that [A 1 ] ik = e t i /T 1,k [ F]km = F(T 1,k, T 2,m) [A 2 ] jm = e t j /T 2,m [Z ] ij = z( t i, t j ) (10) Then Z = A 1 F A T 2 (11) (M 1 M 2 ) = (M 1 K 1 )(K 1 K 2 )(K 2 M 2 )

13 2D NMR Model Define A 1, A 2, F, and Z such that [A 1 ] ik = e t i /T 1,k [ F]km = F(T 1,k, T 2,m) [A 2 ] jm = e t j /T 2,m [Z ] ij = z( t i, t j ) (10) Then Z = A 1 F A T 2 (11) (M 1 M 2 ) = (M 1 K 1 )(K 1 K 2 )(K 2 M 2 ) which becomes vec(z ) = (A 2 A 1 )vec( F) (12)

14 Ordinary Least Squares With 1D ordinary least squares (OLS), we solve min AF y 2 F R K 2 (13)

15 Ordinary Least Squares Define the singular value decomposition of A as A = L i=1 σ i u i v T i (14) where σ i are the singular values and u i, v i are the left and right singular vectors, respectively. Then the OLS solution is F OLS = N i=1 u T i y v i σ i. (15)

16 Regularization To increase stability in the inversion, we add a penalty term tuned by the parameter α. The most common form is Tikhonov regularization, aka ridge regression. min AF y 2 F R K 2 +α2 F 2 2 (16)

17 Regularization Other common penalty forms include: L p regularization, p 1 min AF y 2 F R K 2 +α F p (16) L 1 regularization is known as LASSO. L p regularization for 1 < p < 2 is called bridge regression.

18 Regularization Other common penalty forms include: differential operator, L min F R K AF y 2 2 +α LF 2 2 (16)

19 Regularization We will discuss Tikhonov regularization. min AF y 2 F R K 2 + α2 F 2 2 (16)

20 Regularized Least Squares The regularized problem can be expressed as a normal least squares problem min F R K ÃF ỹ 2 2 (17) where à = ỹ = [ A ] αi K [ ] y 0 K

21 Regularized Least Squares For an appropriate choice of α, F Tikh = N i=1 f i u T i y v i σ i (18) where, in the case of Tikhonov regularization, the filter factors f i take the form f i = σ2 i α 2 + σ 2 i. (19)

22 Regularization Parameter The choice of regularization parameter α strongly influences the character of the solution.

23 Methods to Choose the Regularization Parameter Numerous methods have been proposed to choose the ideal regularization parameter. We will consider the following: Discrepancy Principle Generalized Cross Validation L-Curve Method

24 Discrepancy Principle The discrepancy principle attempts to minimize the residual based on a prespecified error bound ɛ, such that for the optimal α. Disadvantages: AF α y = ɛ (20) Requires a priori knowledge of the error Often oversmooths the solution

25 Generalized Cross Validation Generalized Cross Validation (GCV) extends the idea of leave-one-out cross-validation, minimizing the function with G(α) = AF α y 2 (τ(α)) 2 (21) τ(α) = trace(i A(A T A + α 2 L T L) 1 A T ). (22) where L is a differential operator or the identity matrix (Hansen). Disadvantages: G(α) is difficult to minimize numerically due to its flatness Does not perform well with correlated errors

26 The L-Curve Introduced by P.C. Hansen in 1993, the L-curve was originally used as an analytical tool. It plots the residual AF α y 2 against the size of the solution F α 2 as a function of α.

27 L-Curve Method The L-curve method was proposed by Hansen and O Leary as a means of choosing the regularization parameter α. Idea: Find the corner of the L-curve. Advantages over the other methods: Well-defined numerically Requires no prior knowledge of the errors Not heavily influenced by large correlated errors when considered on a log scale

28 L-Curve Method When GCV performs well, the chosen α value is close to the value chosen by the L-curve method.

29 L-Curve Method (Hansen, O Leary) Let (ρ, η) define a point on the L-curve in log scale. FINDCORNER 1. Calculate several points (ρ i, η i ) on each side of the corner.

30 L-Curve Method (Hansen, O Leary) Let (ρ, η) define a point on the L-curve in log scale. FINDCORNER 2. Fit a 3-dimensional cubic spline to those points (ρ i, η i, α i ) after first performing local smoothing.

31 L-Curve Method (Hansen, O Leary) Let (ρ, η) define a point on the L-curve in log scale. FINDCORNER 3. Compute the point of maximum curvature and find the corresponding regularization parameter α 0.

32 L-Curve Method (Hansen, O Leary) Let (ρ, η) define a point on the L-curve in log scale. FINDCORNER 4. Solve the regularized problem and add the new point (ρ(α 0 ), η(α 0 )) to the L-curve.

33 L-Curve Method (Hansen, O Leary) Let (ρ, η) define a point on the L-curve in log scale. 5. Repeat until convergence. FINDCORNER

34 Model Extensions Characterize the optimal choice of penalty term using the L-curve method to optimize the regularization parameter. L 2 penalty: min F R K AF y α F 2 2

35 Model Extensions Characterize the optimal choice of penalty term using the L-curve method to optimize the regularization parameter. L 2 penalty: min F R K AF y α F 2 2 L 1 penalty: min F R K AF y α F 1

36 Model Extensions Characterize the optimal choice of penalty term using the L-curve method to optimize the regularization parameter. L 2 penalty: min F R K AF y α F 2 2 L 1 penalty: min F R K AF y α F 1 Elastic Net: min F R K AF y α 1 F 1 + α 2 F 2

37 Model Extensions Characterize the optimal choice of penalty term using the L-curve method to optimize the regularization parameter. L 2 penalty: min F R K AF y α F 2 2 L 1 penalty: min F R K AF y α F 1 Elastic Net: min F R K AF y α 1 F 1 + α 2 F 2 L p penalty: min F R K AF y α F p

38 Conclusion An NMR relaxometry signal can be inverted via an inverse Laplace transform. The measurement of an additional, indirect dimension provides increased stability in the inversion. Regularization Form a least-squares problem. Add a Tikhonov regularization term for stability. L-Curve Critical to the quality of the inversion is the choice of regularization parameter α. Use parametric plot of AF α y 2 versus F α 2 to find the optimal α.

39 References I Primary Material: Celik, H., Bouhrara, M., Reiter, D. A., Fishbein, K. W., & Spencer, R. G. (2013). Stabilization of the inverse Laplace transform of multiexponential decay through introduction of a second dimension. Journal of Magnetic Resonance, 236, Hansen, P. C. & O Leary, D. P. (1993). The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems. SIAM Journal on Scientific Computing, 14(6), Secondary Material: Fu, W. J. (1998). Penalized Regressions: The Bridge versus the Lasso. Journal of Computational and Graphical Statistics, 7(3), Varah, J. M. (1983). Pitfalls in the Numerical Solution of Linear Ill-Posed Problems. SIAM Journal on Scientific and Statistical Computing, 4(2),

40 References II Berman, P., Ofer, L., Parmet, Y., Saunders, M., Wiesman, Z. (2013). Laplace Inversion of Low-Resolution NMR Relaxometry Data Using Sparse Representation Methods. Concepts in Magnetic Resonance, 42(3), Hansen, P. C. (1992). Analysis of Discrete Ill-Posed Problems by Means of the L-Curve. SIAM Review, 34(4), Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society, 58(1), Venkataramanan, L., Song, Y., & Hurlimann, M. D., (2002). Solving Fredholm Integrals of the First Kind with Tensor Product Structure in 2 and 2.5 Dimensions. IEEE Transactions on Signal Processing, 50(5),